Question: 2019 Fall Final - 11
Assume you write a multiple-choice exam that has 25 questions. For each question, four options are given
to you, and exactly one of these options is the correct answer. Assume that you answer each question
uniformly at random, where the choices for different questions are independent of each other. What is the
probability that you have exactly 17 correct answers?
(a)
${25 \choose 17} \cdot \left( 1 \middle/ 4 \right)^8 \cdot \left( 3 \middle/ 4 \right)^{17}$
(b)
${25 \choose 17} \cdot \left( 1 \middle/ 4 \right)^{17}$
(c)
${25 \choose 17} \cdot \left( 3 \middle/ 4 \right)^{17}$
(d)
${25 \choose 17} \cdot \left( 1 \middle/ 4 \right)^{17} \cdot \left( 3 \middle/ 4 \right)^8$